Download Slides 11: Beyond Ito Processes (PDF, 130 KB)

Stochastic Models
Non-Ito Stochastic Processes
Walt Pohl
Universität Zürich
Department of Business Administration
April 24, 2013
Beyond Ito Processes
We have shown that the class of Ito processes is hard to
escape:
Simulating a process using any (finite-variance)
distribution for steps gives you an Ito process.
Any nonlinear function of an Ito process is an Ito
process.
Does this mean that stochastic processes are all Ito
processes? Far from it.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
2 / 15
Recall Wiener Process
The Wiener process satisfies three properties.
W0 = 0
Wt − Wt0 ∼ Normal(0, t − t 0 ).
For t1 < t2 < t10 < t20 , Wt2 − Wt1 and Wt20 − Wt10 are
independent.
Can we replace the distribution in property 2?
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
3 / 15
Infinitely Divisible Distributions
We need to be able to write the increments as sums of
smaller increments:
Xt3 − Xt1 = (Xt3 − Xt2 ) + (Xt2 − Xt1 )
A distribution that can be written as a sum of n
distributions is called infinitely divisible.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
4 / 15
Example: Normal Distribution
A normal random variable X ∼ Normal(µ, σ 2 ) can be
written as a sum
N
X
X =
Xi ,
i=1
where Xi ∼ Normal(µ/N, σ 2 /N).
But is it the only one?
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
5 / 15
Example: Poisson Distribution
The Poisson distribution Poisson(λ) is an integer-valued
random variable N, such that
P(X = n) = e
X satisfies
X =
N
X
n
−λ λ
n!
.
Xi ,
i=1
where Xi ∼ Poisson(λ/N).
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
6 / 15
Example: Poisson Distribution, cont’d
Note that (unlike the p
normal case) Poisson(λ/N) is not
Poisson(λ) scaled by (1/N).
Poisson(λ/N) takes on integer values.
p
Poisson(X ) scaled by (1/N) takes on values of
the form
n
√
N
for integer n.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
7 / 15
The Poisson Process
The Poisson process is the unique continuous-time
process such that:
N0 = 0
Nt − Wt0 ∼ Poisson(λ(t − t 0 )).
For t1 < t2 < t10 < t20 , Nt2 − Nt1 and Nt20 − Nt10 are
independent.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
8 / 15
Simulating the Poisson Process
Simulating the Poisson process is straightforward.
Choose a time step, ∆t, and then simulate Nt+∆t as
Nt+∆t = Nt + q,
where q ∼ Poisson(λ∆t).
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
9 / 15
Simulating the Poisson Process, cont’d
Note that for very small time intervals, having two
events occur in the same interval becomes very unlikely,
and Poisson(λ∆t) can be approximated by a coin flip.
P(N = 0) = e −λ∆t ≈ 1 − λ∆t
P(N = 1) = λe −λ∆t ≈ λ∆t
So we can simulate the Poisson process via coin flips.
We can also write downa binomial tree model, and
extend our derivative pricing results. But the Poisson
process is rather special.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
10 / 15
Jump Process
A Poisson process is an example of a (pure) jump
process: the process changes level is discrete jumps.
Logically, we can split the idea of a jump process into
two pieces:
1
2
At a given instance, does the process jump?
Given that it jumped, by how much?
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
11 / 15
Compound Poisson Process
A compound Poisson process is X is given by two
parameters, λ, and a jump distribution J.
X0 = 0
The number of jumps in an interval is given by
Poisson(λ(t − t 0 )). Each jump is distributed J. (So
if n jumps occur in an interval, then
Xt − Xt 0 =
n
X
Yi ,
i=1
where each Yi has distribution J.
For t1 < t2 < t10 < t20 , Nt2 − Nt1 and Nt20 − Nt10 are
independent.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
12 / 15
The Compound Poisson Process and Coin
Flips
The definition gives a reciple for simulating the
compound Poisson process. Notice one key difference
with the regular Poisson process.
For small ∆t, the probability of a jump occurring
remains a coin flip, with probability λ∆t, but since
jumps of different sizes can occur, we can no longer
write down a binomial tree.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
13 / 15
Jump-Diffusion Model
Stock prices don’t move exclusively in jumps – they
feature many small movements, and the occasional big
movement. In a jump-diffusion model, we allow both
possibilities, by adding together a geometric Brownian
motion and a compound Poisson process. This is
normally written:
dS/S = µdt + σdW + Ydq.
The extra term is a compound Poisson process with
jump distribution Y and parameter λ (not shown).
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
14 / 15
But How Do We Price?
We can hedge the small movements, but not the big
ones. How do we write down the price? We’re back in
the realm of decision theory: what are the jumps worth
to us?
One common answer is Merton’s – the value of the jump
is its expected value.
The economic rationale is that maybe each jump is
idiosyncratic to an individual stock, so that you can
hedge them by holding a diversified portfolio. This only
works if stocks usually don’t jump together. (But
sometimes they do...)
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
15 / 15